4.1 Pressure & Its measurement (Page 3/14)

Ut austin - principles of Page 3 / 14

If the liquid is water, normal atmospheric pressure will support a column of water over 10 meters high, which is rather inconvenient for making (and reading) a barometer. Because mercury (Hg) is about 13.6-times denser than water, a mercury barometer only needs to be $\frac{1}{13.6}$ as tall as a water barometer—a more suitable size. Standard atmospheric pressure of 1 atm at sea level (101,325 Pa) corresponds to a column of mercury that is about 760 mm (29.92 in.) high. The torr was originally intended to be a unit equal to one millimeter of mercury, but it no longer corresponds exactly. The pressure exerted by a fluid due to gravity is known as hydrostatic pressure , p :

p = h ρ g

where h is the height of the fluid, ρ is the density of the fluid, and g is acceleration due to gravity.

Calculation of barometric pressure

Show the calculation supporting the claim that atmospheric pressure near sea level corresponds to the pressure exerted by a column of mercury that is about 760 mm high. The density of mercury = 13.6 g/cm ³ .

Solution

The hydrostatic pressure is given by p = hρg , with h = 760 mm, ρ = 13.6 g/cm ³ , and g = 9.81 m/s ² . Plugging these values into the equation and doing the necessary unit conversions will give us the value we seek. (Note: We are expecting to find a pressure of ~101,325 Pa:)

101,325 N {/m}^{2} = 101,325 \frac{{kg·m/s}^{2}}{m^{2}} = 101,325 \frac{kg}{{m·s}^{2}}

p = (760 mm \times \frac{1 m}{1000 mm}) \times (\frac{13.6 g}{1 {cm}^{3}} \times \frac{1 kg}{1000 g} \times \frac{{(100 cm)}^{3}}{{(1 m)}^{3}}) \times (\frac{9.81 m}{1 s^{2}})

= (0.760 m) (13,600 {kg/m}^{3}) (9.81 {m/s}^{2}) = 1.01 \times 10^{5} {kg/ms}^{2} = 1.01 \times 10^{5} {N /m}^{2}

= 1.01 \times 10^{5} Pa

Check your learning

Calculate the height of a column of water at 25 °C that corresponds to normal atmospheric pressure. The density of water at this temperature is 1.0 g/cm ³ .

Answer:

10.3 m

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A manometer is a device similar to a barometer that can be used to measure the pressure of a gas trapped in a container. A closed-end manometer is a U-shaped tube with one closed arm, one arm that connects to the gas to be measured, and a nonvolatile liquid (usually mercury) in between. As with a barometer, the distance between the liquid levels in the two arms of the tube ( h in the diagram) is proportional to the pressure of the gas in the container. An open-end manometer ( [link] ) is the same as a closed-end manometer, but one of its arms is open to the atmosphere. In this case, the distance between the liquid levels corresponds to the difference in pressure between the gas in the container and the atmosphere.

Three diagrams of manometers are shown. Each manometer consists of a spherical pink container filled with gas on the left that is connected to a U-shaped, sealed tube by a valve on its right. The top of the U-shape aligns with the gas-filled sphere and the U, which extends below, contains mercury. The first manometer has a sealed tube. The sealed end to the upper right in the diagram is labeled “closed end” and “vacuum.” An arrow points downward in the left side of the U shaped tube to the mercury surface. This arrow is labeled “P subscript gas.” The mercury level is higher in the right side of the tube than in the left. The difference in height is labeled “h.” Beneath this manometer illustration appears the label P subscript gas equal sign h rho g. The second manometer has an open ended tube, which is labeled “open end.” At this opening in the upper right of the diagram is a downward arrow, above which is the label P subscript a t m. An arrow points downward in the left side of the U shaped tube to the mercury surface. This arrow is labeled “P subscript gas.” The mercury level is higher in the left side of the tube than in the right. This difference in height is labeled “h.” Beneath this manometer illustration appears the label P subscript gas equal sign P subscript a t m minus sign h rho g. The third manometer has an open ended tube and is similar to the second manometer except that mercury level is higher in the right side of the tube than in the left. This difference in height is labeled “h.” Beneath this manometer illustration appears the label P subscript gas equal sign P subscript a t m plus h rho g. — A manometer can be used to measure the pressure of a gas. The (difference in) height between the liquid levels ( h ) is a measure of the pressure. Mercury is usually used because of its large density.

Calculation of pressure using a closed-end manometer

The pressure of a sample of gas is measured with a closed-end manometer, as shown to the right. The liquid in the manometer is mercury. Determine the pressure of the gas in:

(a) torr

(b) Pa

A diagram of a closed-end manometer is shown. To the upper left is a spherical container labeled, “gas.” This container is connected by a valve to a U-shaped tube which is labeled “closed end” at the upper right end. The container and a portion of tube that follows are shaded pink. The lower portion of the U-shaped tube is shaded grey with the height of the gray region being greater on the right side than on the left. The difference in height between the left side and right side is 26.4 c m which is indicated with horizontal line segments and arrows.

Solution

The pressure of the gas is equal to a column of mercury of height 26.4 cm. (The pressure at the bottom horizontal line is equal on both sides of the tube. The pressure on the left is due to the gas and the pressure on the right is due to 26.4 cm Hg, or mercury.) We could use the equation p = hρg as in [link] , but it is simpler to just convert between units using [link] .

(a) $26.4 cm Hg \times \frac{10 mm Hg}{1 cm Hg} \times \frac{1 torr}{1 mm Hg} = 264 torr$

(b) $264 torr \times \frac{1 atm}{760 torr} \times \frac{101,325 Pa}{1 atm} = 35,200 Pa$

Check your learning

The pressure of a sample of gas is measured with a closed-end manometer. The liquid in the manometer is mercury. Determine the pressure of the gas in:

(a) torr

(b) Pa

Answer:

(a) ~150 torr; (b) ~20,000 Pa; (c) ~0.20 bar

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Source: OpenStax, Ut austin - principles of chemistry. OpenStax CNX. Mar 31, 2016 Download for free at http://legacy.cnx.org/content/col11830/1.13

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